It is well known that the compactly supported wavelets cannot belong to
the class $C^\infty({\bf R})\cap L^2({\bf R})$. This is also true for
wavelets with exponential decay. We show that one can construct
wavelets in the class $C^\infty({\bf R})\cap L^2({\bf R})$ that are
``almost'' of exponential decay and, moreover, they are
band-limited. We do this by showing that we can adapt the
construction of the Lemari\'e-Meyer wavelets \cite{LM} that
is found in \cite{BSW} so that we obtain band-limited,
$C^\infty$-wavelets on $\bf R$ that have subexponential decay,
that is, for every $00$
such that $|\psi(x)|\leq C_\varepsilon e^{-|x|^{1-\varepsilon}}$,
$x\in\bf R$. Moreover, all of its derivatives have also
subexponential decay. The proof is constructive and uses the
Gevrey classes of functions.